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Some nonexistence theorems for semilinear fourth-order equations

Part of: Partial differential equations

Published online by Cambridge University Press:  27 December 2018

M. Á. Burgos-Pérez ,
J. García-Melián  and
A. Quaas
M. Á. Burgos-Pérez
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38200 La Laguna, Spain (miguelburgosperez@gmail.com)
J. García-Melián
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38200 La Laguna, Spainand Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica, Molecular y Fotónica, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38200 La Laguna, Spain (jjgarmel@ull.es)
A. Quaas
Affiliation:
Departamento de Matemática,Universidad Técnica Federico Santa María, Casilla V-110, Avda. España, 1680 Valparaíso, Chile (alexander.quaas@usm.cl)
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Abstract

In this paper, we analyse the semilinear fourth-order problem ( − Δ)2u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and

$$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$
for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu > 0 in ℝN, and they do not exist when the previous condition fails.

Keywords

Liouville theoremsmaximum principleelliptic systems

MSC classification

Primary: 35B53: Liouville theorems, Phragmén-Lindelöf theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 761 - 779
Copyright
Copyright © Royal Society of Edinburgh 2018 

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